---
tag: 0QRQVH0CRW
kind: definition
name: Upper and lower Darboux sums
msc: [26A42]
defines: 0K0A06X2MQ
uses: [069P20KYQT, 05FQ3VGAR9]
sources:
  - url: https://en.wikipedia.org/wiki/Darboux_integral
    locator: Darboux sums
---

## Definition

Let $a < b$, let $f \colon [a, b] \to \mathbb{R}$ be bounded, and let $P = (x_0, \dots, x_n)$ be a [[partition|069P20KYQT]] of $[a, b]$.
For each $i$, let $M_i = \sup f([x_{i-1}, x_i])$ and $m_i = \inf f([x_{i-1}, x_i])$.
The upper and lower Darboux sums of $f$ for $P$ are
$$U(f, P) = \sum_{i=1}^{n} M_i (x_i - x_{i-1}), \qquad L(f, P) = \sum_{i=1}^{n} m_i (x_i - x_{i-1}).$$

## Remarks

Always $(b - a) \inf f \le L(f, P) \le U(f, P) \le (b - a) \sup f$.
For a [[refinement|0EEKF7YFBW]] $Q$ of $P$, we have $L(f, P) \le L(f, Q)$ and $U(f, Q) \le U(f, P)$.
So $L(f, P) \le U(f, Q)$ for any two partitions $P$ and $Q$.

## Explanations

### L1 human

The upper sum covers the graph with rectangles that reach its highest point on each piece.
The lower sum uses rectangles that reach its lowest point on each piece.